A245595 Unique integer r with -prime(n)/2 < r <= prime(n)/2 such that S(n) == r (mod prime(n)), where S(n) is the large Schroeder number A006318(n).

0, 0, 2, -1, -2, -1, 7, -5, -5, 11, 10, -11, 11, 12, 2, 17, -2, 19, -15, -26, 33, 17, -22, -11, 18, 8, 18, -27, 17, 51, -37, -34, 28, -4, 66, -37, -69, -58, 45, -81, -20, -86, -19, 17, -12, -30, 35, -32, 5, -11, -8, -45, 12, -111, -28, -71, 76, 59, 102, -25

Offset: 1

Author

Zhi-Wei Sun, Jul 27 2014

Keywords

• sign

Comments

Conjecture: (i) For any integer n > 2, the term a(n) is nonzero, i.e., prime(n) does not divide the large Schroeder number S(n).

(ii) For any integer n > 2, prime(n) does not divide the Bell number B(2*n) = A000110(2*n).

We have verified parts (i) and (ii) for n up to 440000 and 66000 respectively.

Conjecture (i) fails for the first time for n=20239789. In particular, a(20239789)=0. - Max Alekseyev, Oct 05 2015

Examples

			a(5) = -2 since S(5) = 394 == -2 (mod prime(5)=11).
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A000040, A000110, A006318, A245525.

Programs

Mathematica

rMod[m_,n_]:=Mod[m,n,-(n-1)/2]
S[n_]:=Sum[Binomial[n+k,2k]*Binomial[2k,k]/(k+1),{k,0,n}]
a[n_]:=rMod[S[n],Prime[n]]
Table[a[n],{n,1,60}]